Music theory

Music theory is the study of the practices and possibilities of music. It generally derives from observation of how musicians and composers make music, but includes hypothetical speculation. Most commonly, the term describes the academic study and analysis of fundamental elements of music such as pitch, rhythm, harmony, and form, but also refers to descriptions, concepts, or beliefs related to music. Because of the ever-expanding conception of what constitutes music (see Definition of music), a more inclusive definition could be that music theory is the consideration of any sonic phenomena, including silence, as it relates to music.

Music theory is a subfield of musicology, which is itself a subfield within the overarching field of the arts and humanities. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a spectacle.[1] As such, it is often concerned with abstract musical aspects such as tuning and tonal systems, scales, consonance and dissonance, and rhythmic relationships, but there is also a body of theory concerning such practical aspects as the creation or the performance of music, orchestration, ornamentation, improvisation, and electronic sound production.[2] A person working in music theory is a music theorist. Methods of analysis include mathematics, graphic analysis, and, especially, analysis enabled by Western music notation. Comparative, descriptive, statistical, and other methods are also used.

The development, preservation, and transmission of music theory may be found in oral and practical music-making traditions, musical instruments, and other artifacts. For example, ancient instruments from Mesopotamia, China,[3] and prehistoric sites around the world reveal details about the music they produced and, potentially, something of the musical theory that might have been used by their makers (see History of music and Musical instrument). In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia, Pharoanic Egypt, and ancient China have also considered music theory in more formal ways such as written treatises and music notation.

Ancient instruments, artifacts, and later, depictions of performance in artworks give insight into early music-making. As early as the Paleolithic era, it appears people considered elements of music in some way. For instance, a bone flute with carefully placed finger holes found in Hohle Fels in Germany and dated c. 35,000 BCE,[4] may be a prehistoric example of the manufacture of an instrument to produce a preconceived set of pitches. For further discussion of Upper Paleolithic flutes, see d'Errico, et al. 2003, 39–48.

Similar bone flutes (gǔdí, 贾湖骨笛) from Neolithic Jiahu, China dated c. 7,000 BCE[5] reveal their makers progressively added more holes to expand their scales, structured pitch intervals closer to each other to adjust tuning, and could play increasingly expressive and varied music.[6] "Tonal analysis of the flutes revealed that the seven holes [in some of the flutes] correspond to a tone scale remarkably similar to Western eight-pitch scales."[7][8] These instruments[9] indicate their makers became familiar with acoustics and developed theories of music comparable to those of later times. Audio recordings of two of these flutes by Brookhaven National Laboratory are available here.

In North America, similar flutes from the Anasazi Indian culture were found in Arizona and dated c. 600–750 CE, but again, suggest an older tradition. These instruments typically have six finger holes ranging one and a half octaves.[10] As with all these ancient flutes, it is likely an error to imagine the Anasazi flutes were limited to only as many tones as they have holes. Changes in embouchure, overblowing, and cross-fingering are common techniques on modern flutes like these that produce a much larger range of notes within an octave and in octaves above the fundamental octave.[11]

The earliest known examples of written music theory are inscribed on clay tablets found in Iraq and Syria, some of which contain lists of intervals and other details[12] from which "...musicologists have been able to produce credible reconstructions of the Mesopotamian tonal and tuning systems."[13] Tablets from Ugarit contain what are known as the Hurrian songs or Hurrian Hymns dated c. 1,400 BCE. An interpretation of the only substantially complete Hurrian Hymn, h.6, may be heard here. The system of phonetic notation in Sumer and Babylonia is based on a music terminology that gives individual names to nine musical strings or "notes", and to fourteen basic terms describing intervals of the fourth and fifth that were used in tuning string instruments (according to seven heptatonic diatonic scales), and terms for thirds and sixths that appear to have been used to fine-tune (or temper in some way) the seven notes generated for each scale.[14][15][16][17][18]

Over time, many cultures began to record their theories of music in writing by describing practices and theory that was previously developed and passed along through oral tradition. In cultures where no written examples exist, oral traditions indicate a long history of theoretical consideration, often with unique concepts of use, performance, tuning and intervals, and other fundamental elements of music. The Vedas, the sacred texts of India (c. 1,000–500 BCE) contain theoretical discussion of music in the Sama Veda and Yajur-Veda. The Natya Shastra,[19] written between 200 BCE to 200 CE and attributed to Bharata Muni, discusses classes of melodic structure, intervals, consonance and dissonance, performance, and other theoretical aspects such a "shruti," defined as the least perceivable difference between two pitches.[20]

The music of pre-Columbian Mesoamerica is known through the many instruments discovered. Thirty-two condor-bone flutes and thirty-seven cornet-like instruments made of deer and llama bones have been recovered from a site at Caral, Peru dating to c. 2,100 BCE.[21][22][23] Flute No. 15 produces five distinct fundamental tones. A Mayan marimba-like instrument (c. 350 CE), made from five turtle shells of decreasing sizes suspended on a wooden frame, has been discovered in Belize.[24]

Later artwork depicts ensemble and solo performance. Taken together, this evidence does not in itself demonstrate anything about music theory in Mesoamerica from at least 2,000 BCE, though "...it is widely accepted that finds and depictions of ancient musical instruments are not only markers of musical traditions in space and time. … The information obtained from the archaeological record can be deepened considerably when ancient scripts, historical treaties, and other written sources concerning music are related. Such documents offer notes on performance practices and their sociocultural contexts. For some cultures, hints concerning ancient music theory and musical aesthetics may also be found."[25]

Music theory in ancient Africa can also be seen in instruments .[26] The Mbira, a wood or bamboo-tined instrument similar to a Kalimba, appeared on the west coast of Africa about 3,000 years ago, and metal-tined lamellophones appeared in the Zambezi River valley around 1,300 years ago.[27] In the 20th century, these instruments produce a number of tones, ranging to 32 separate pitches, and demonstrate a great variety of tunings—tunings "so dissimilar as to offer no apparent common foundation", something that might have been expected at least by 1932.[28] The djembé, a common type of drum, likely originated from earlier, extremely ancient drums.[29] Djembé ensembles create complex polyrhythmic patterns,[30] but produce a variety of pitches depending on size and playing technique, usually producing at least three separate tones.[31] African music theory is also preserved in oral and cultural traditions that are one example of the great variety of concepts of fundamental aspects of music around the world.[32]

In China, a variety of wind, string, percussion instruments, and written descriptions and drawings of them from the Shang Dynasty (c. 16th to 11th century BCE), show sophisticated form and design.[33] During the Zhou Dynasty (c. 1046–256 BC), a formal system of court and ceremonial music later termed "yayue" was established. As early as the 7th century BCE, a system of pitch generation was described based on a ratio of 2:3 and a pentatonic scale was derived from the cycle of fifths,[34] the beginnings of which appear in 7,000 year-old Jiahu bone flutes. In the tomb of the Marquis Yi of Zeng (5th century BCE), among many other instruments, a set of bronze chime bells were found that sound five complete seven note octaves in the key of C Major and include twelve semitones.[35] The Analects of Confucius, believed to have been written c. 475 to 221 BC, discuss the aesthetics of what Confucius considered the most benevolent form and use of music, in contrast to popular music of his time—an example of early music criticism and consideration of aesthetics.[36][37]

A set of bells from China, 5th Century BCE.

Around the time of Confucius, the ancient Greeks, notably Pythagoras (c. 530 BCE), Aristotle (c. 350 BCE),[38] Aristoxenus (c. 335 BCE),[39] and later Ptolemy (c. 120 CE),[40] speculated and experimented with ideas that became the basis of music theory in Middle Eastern and Western cultures during the Middle Ages as can be seen, for example, in the writing of Boethius in 5th century Rome[41] and Yunus al-Katibin 7th century Medina.[42] Middle Eastern and Western theory diverged in different directions from ancient Greek theory and created what are now two distinctly different bodies of theory and styles of music.

As Western musical influence spread throughout the world in the 1800s, musicians adopted Western theory as an international standard—but other theoretical traditions in both textual and oral traditions remain in use. For example, the long and rich musical traditions unique to ancient and current cultures of Africa are primarily oral, but describe specific forms, genres, performance practices, tunings, and other aspects of music theory.[43][44]

Pitch is the lowness or highness of a tone, for example the difference between middle C and a higher C. The frequency of the sound waves producing a pitch can be measured precisely, but the perception of pitch is more complex because we rarely hear a single frequency or pure pitch. In music, tones, even those sounded by solo instruments or voices, are usually a complex combination of frequencies, and therefore a mix of pitches. Accordingly, theorists often describe pitch as a subjective sensation.[46]

Most people appear to possess relative pitch, which means they perceive each note relative to some reference pitch, or as some interval from the previous pitch. Significantly fewer people demonstrate absolute pitch (or perfect pitch), the ability to identify pitches without comparison to another pitch. Human perception of pitch can be comprehensively fooled to create auditory illusions. Despite these perceptual oddities, perceived pitch is nearly always closely connected with the fundamental frequency of a note, with a lesser connection to sound pressure level, harmonic content (complexity) of the sound, and to the immediately preceding history of notes heard.[47] In general, the higher the frequency of vibration, the higher the perceived pitch. The lower the frequency, the lower the pitch.[48] However, even for tones of equal intensity, perceived pitch and measured frequency do not stand in a simple linear relationship.[49]

Intensity (loudness) can change perception of pitch. Below about 1000 Hz, perceived pitch gets lower as intensity increases. Between 1000 and 2000 Hz, pitch remains fairly constant. Above 2000 Hz, pitch rises with intensity.[50] This is due to the ear's natural sensitivity to higher pitched sound, as well as the ear's particular sensitivity to sound around the 2000–5000 Hz interval,[51] the frequency range most of the human voice occupies.[52]

The difference in frequency between two pitches is called an interval. The most basic interval is the unison, which is simply two notes of the same pitch, followed by the slightly more complex octave: pitches that are either double or half the frequency of the other. The unique characteristics of octaves gave rise to the concept of what is called pitch class, an important aspect of music theory. Pitches of the same letter name that occur in different octaves may be grouped into a single "class" by ignoring the difference in octave. For example, a high C and a low C are members of the same pitch class—the class that contains all C's. The concept of pitch class greatly aids aspects of analysis and composition.[53]

Although pitch can be identified by specific frequency, the letter names assigned to pitches are somewhat arbitrary. For example, today most orchestras assign Concert A (the A above middle C on the piano) to the specific frequency of 440 Hz, rather than, for instance, 435Hz as it was in France in 1859. In England, that A varied between 439 and 452. These differences can have a noticeable effect on the timbre of instruments and other phenomena. Many cultures do not attempt to standardize pitch, often considering that it should be allowed to vary depending on genre, style, mood, etc. In historically informed performance of older music, tuning is often set to match the tuning used in the period when it was written. A frequency of 440 Hz was recommended as the standard pitch for Concert A in 1939, and in 1955 the International Organization for Standardization affirmed the choice.[54]A440 is now widely, though not exclusively, the standard for music around the world.

Pitch is also an important consideration in tuning systems, or temperament, used to determine the intervallic distance between tones, as within a scale. Tuning systems vary widely within and between world cultures. In Western culture, there have long been several competing tuning systems, all with different qualities. Internationally, the system known as equal temperament is most commonly used today because it is considered the most satisfactory compromise that allows instruments of fixed tuning (e.g. the piano) to sound acceptably in tune in all keys.

A pattern of whole and half steps in the Ionian mode or major scale on C Play(help·info).

Notes can be arranged in a variety of scales and modes. Western music theory generally divides the octave into a series of twelve tones, called a chromatic scale, within which the interval between adjacent tones is called a half step or semitone. In equal temperament each semitone is equidistant from the next, but other tuning systems are also used. Selecting tones from this set of 12 and arranging them in patterns of semitones and whole tones creates other scales.[55]

The most commonly encountered scales are the seven-toned major, the harmonic minor, the melodic minor, and the natural minor. Other examples of scales are the octatonic scale and the pentatonic or five-tone scale, which is common in folk music and blues. Non-Western cultures often use scales that do not correspond with an equally divided twelve-tone division of the octave. For example, classical Ottoman, Persian, Indian and Arabic musical systems often make use of multiples of quarter tones (half the size of a semitone, as the name indicates), for instance in 'neutral' seconds (three quarter tones) or 'neutral' thirds (seven quarter tones)—they do not normally use the quarter tone itself as a direct interval.[56]

In traditional Western notation, the scale used for a composition is usually indicated by a key signature at the beginning to designate the pitches that make up that scale. As the music progresses, the pitches used may change and introduce a different scale. Music can be transposed from one scale to another for various purposes, often to accommodate the range of a vocalist. Such transposition raises or lowers the overall pitch range, but preserves the intervallic relationships of the original scale. For example, transposition from the key of C major to D major raises all pitches of the scale of C major equally by a whole tone. Since the interval relationships remain unchanged, transposition may be unnoticed by a listener, however other qualities may change noticeably because transposition changes the relationship of the overall pitch range compared to the range of the instruments or voices that perform the music. This often affects the music's overall sound, as well as having technical implications for the performers.[57]

The interrelationship of the keys most commonly used in Western tonal music is conveniently shown by the circle of fifths. Unique key signatures are also sometimes devised for a particular composition. During the Baroque period, emotional associations with specific keys, known as the doctrine of the affections, were an important topic in music theory, but the unique tonal colorings of keys that gave rise to that doctrine were largely erased with the adoption of equal temperament. However, many musicians continue to feel that certain keys are more appropriate to certain emotions than others. Indian classical music theory continues to strongly associate keys with emotional states, times of day, and other extra-musical concepts and notably, does not employ equal temperament.

Consonance and dissonance are subjective qualities of the sonority of intervals that vary widely in different cultures and over the ages. Consonance (or concord) is the quality of an interval or chord that seems stable and complete in itself. Dissonance (or discord) is the opposite in that it feels incomplete and "wants to" resolve to a consonant interval. Dissonant intervals seem to clash. Consonant intervals seem to sound comfortable together. Commonly, perfect fourths, fifths, and octaves and all major and minor thirds and sixths are considered consonant. All others are dissonant to greater or lesser degree.[58]

Context and many other aspects can affect apparent dissonance and consonance. For example, in a Debussy prelude, a major second may sound stable and consonant, while the same interval may sound dissonant in a Bach fugue. In the Common Practice era, the perfect fourth is considered dissonant when not supported by a lower third or fifth. Since the early 20th century, Arnold Schoenberg’s concept of "emancipated" dissonance, in which traditionally dissonant intervals can be treated as "higher," more remote consonances, has become more widely accepted.[59]

Dissonance is an essential element of music and used in most every culture and genre, not only for effect, but as a fundamental structural element to create motion and tension. J.S. Bach’s music depends in great part on the effect of dissonance. The art of melody writing depends heavily upon the selection of consonant and dissonant tones.[citation needed]

Rhythm is produced by the sequential arrangement of sounds and silences in time. Meter measures music in regular pulse groupings, called measures or bars. The time signature or meter signature specifies how many beats are in a measure, and which value of written note is counted or felt as a single beat.

Through increased stress, or variations in duration or articulation, particular tones may be accented. There are conventions in most musical traditions for regular and hierarchical accentuation of beats to reinforce a given meter. Syncopated rhythms contradict those conventions by accenting unexpected parts of the beat. Playing simultaneous rhythms in more than one time signature is called polymeter. See also polyrhythm.

A melody is a series of tones sounding in succession that typically move toward a climax of tension then resolve to a state of rest. Because melody is such a prominent aspect in so much music, its construction and other qualities are a primary interest of music theory.

The basic elements of melody are pitch, duration, rhythm, and tempo. The tones of a melody are usually drawn from pitch systems such as scales or modes. Melody may consist, to increasing degree, of the figure, motive, semi-phrase, antecedent and consequent phrase, and period or sentence. The period may be considered the complete melody, however some examples combine two periods, or use other combinations of constituents to create larger form melodies.[61]

A chord, in music, is any harmonic set of three or more notes that is heard as if sounding simultaneously.[62][63] These need not actually be played together: arpeggios and broken chords may, for many practical and theoretical purposes, constitute chords. Chords and sequences of chords are frequently used in modern Western, West African.[64] and Oceanian[65] music, whereas they are absent from the music of many other parts of the world.[66]

The most frequently encountered chords are triads, so called because they consist of three distinct notes: further notes may be added to give seventh chords, extended chords, or added tone chords. The most common chords are the major and minortriads and then the augmented and diminishedtriads. The descriptions major, minor, augmented, and diminished are sometimes referred to collectively as chordal quality. Chords are also commonly classed by their root note—so, for instance, the chord C major may be described as a triad of major quality built upon the note C. Chords may also be classified by inversion, the order in which the notes are stacked.

In music, harmony is the use of simultaneous pitches (tones, notes), or chords.[66] The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them.[69] Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic line, or the "horizontal" aspect.[70]Counterpoint, which refers to the interweaving of melodic lines, and polyphony, which refers to the relationship of separate independent voices, are thus sometimes distinguished from harmony.[citation needed]

In popular and jazz harmony, chords are named by their root plus various terms and characters indicating their qualities. For example, a lead sheet may indicate chords such as C major, D minor, and G dominant seventh. In many types of music, notably Baroque, Romantic, modern, and jazz, chords are often augmented with "tensions". A tension is an additional chord member that creates a relatively dissonant interval in relation to the bass. Typically, in the classical common practice period a dissonant chord (chord with tension) "resolves" to a consonant chord. Harmonization usually sounds pleasant to the ear when there is a balance between the consonant and dissonant sounds. In simple words, that occurs when there is a balance between "tense" and "relaxed" moments.[citation needed]

Spectrogram of the first second of an E9 chord played on a Fender Stratocaster guitar with noiseless pickups. Below is the E9 chord audio:

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Timbre, sometimes called "color", or "tone color," is the principal phenomenon that allows us to distinguish one instrument from another when both play at the same pitch and volume, a quality of a voice or instrument often described in terms like bright, dull, shrill, etc. It is of considerable interest in music theory, especially because it is one component of music that has as yet, no standardized nomenclature. It has been called "...the psychoacoustician's multidimensional waste-basket category for everything that cannot be labeled pitch or loudness,"[71] but can be accurately described and analyzed by Fourier analysis and other methods[72] because it results from the combination of all sound frequencies, attack and release envelopes, and other qualities that comprise a tone.

Timbre is principally determined by two things: (1) the relative balance of overtones produced by a given instrument due its construction (e.g. shape, material), and (2) the envelope of the sound (including changes in the overtone structure over time). Timbre varies widely between different instruments, voices, and to lesser degree, between instruments of the same type due to variations in their construction, and significantly, the performer's technique. The timbre of most instruments can be changed by employing different techniques while playing. For example, the timbre of a trumpet changes when a mute is inserted into the bell, the player changes their embouchure, or volume.[citation needed]

A voice can change its timbre by the way the performer manipulates their vocal apparatus, (e.g. the shape of the vocal cavity or mouth). Musical notation frequently specifies alteration in timbre by changes in sounding technique, volume, accent, and other means. These are indicated variously by symbolic and verbal instruction. For example, the word dolce (sweetly) indicates a non-specific, but commonly understood soft and "sweet" timbre. Sul tasto instructs a string player to bow near or over the fingerboard to produce a less brilliant sound. Cuivre instructs a brass player to produce a forced and stridently brassy sound. Accent symbols like marcato (^) and dynamic indications (pp) can also indicate changes in timbre.[citation needed]

In music, "dynamics" normally refers to variations of intensity or volume, as may be measured by physicists and audio engineers in decibels or phons. In music notation, however, dynamics are not treated as absolute values, but as relative ones. Because they are usually measured subjectively, there are factors besides amplitude that affect the performance or perception of intensity, such as timbre, vibrato, and articulation.

The conventional indications of dynamics are abbreviations for Italian words like forte (f) for loud and piano (p) for soft. These two basic notations are modified by indications including mezzo piano (mp) for moderately soft (literally "half soft") and mezzo forte (mf) for moderately loud, sforzando or sforzato (sfz) for a surging or "pushed" attack, or fortepiano (fp) for a loud attack with a sudden decrease to a soft level. The full span of these markings usually range from a nearly inaudible pianissississimo (pppp) to a loud-as-possible fortissississimo (ffff).

Greater extremes of pppppp and fffff and nuances such as p+ or più piano are sometimes found. Other systems of indicating volume are also used in both notation and analysis: dB (decibels), numerical scales, colored or different sized notes, words in languages other than Italian, and symbols such as those for progressively increasing volume (crescendo) or decreasing volume (decrescendo), often called "hairpins" when indicated with diverging or converging lines as shown in the graphic above.

Articulation is the manner in which the performer sounds notes. For example, staccato is the shortening of duration compared to the written note value, legato performs the notes in a smoothly joined sequence with no separation. Articulation is often described rather than quantified, therefore there is room to interpret how to execute precisely each articulation.

For example, staccato is often referred to as "separated" or "detached" rather than having a defined or numbered amount by which to reduce the notated duration. Violin players use a variety of techniques to perform different qualities of staccato. The manner in which a performer decides to execute a given articulation is usually based on the context of the piece or phrase, but many articulation symbols and verbal instructions depend on the instrument and musical period (e.g. viol, wind; classical, baroque; etc.).

There are a set of articulations that most all instruments and voices perform in common. They are, in order of long to short: legato (smooth, connected); tenuto (pressed or played to full notated duration); marcato (accented and detached); staccato ("separated", "detached"); martelé (heavily accented or "hammered"). Many of these can be combined to create certain "in-between" articulations. For example, portato is the combination of tenuto and staccato. Some instruments have unique methods by which to produce sounds, such as spicatto for bowed strings, where the bow bounces off the string.

In music, texture is how the melodic, rhythmic, and harmonic materials are combined in a composition, thus determining the overall quality of the sound in a piece. Texture is often described in regard to the density, or thickness, and range, or width, between lowest and highest pitches, in relative terms as well as more specifically distinguished according to the number of voices, or parts, and the relationship between these voices. For example, a thick texture contains many "layers" of instruments. One of these layers could be a string section, or another brass.

The thickness also is affected by the amount and the richness of the instruments playing the piece. The thickness varies from light to thick. A lightly textured piece will have light, sparse scoring. A thickly or heavily textured piece will be scored for many instruments. A piece's texture may be affected by the number and character of parts playing at once, the timbre of the instruments or voices playing these parts and the harmony, tempo, and rhythms used.[74] The types categorized by number and relationship of parts are analyzed and determined through the labeling of primary textural elements: primary melody, secondary melody, parallel supporting melody, static support, harmonic support, rhythmic support, and harmonic and rhythmic support.[75]

Common types included monophonic texture (a single melodic voice, such as a piece for solo soprano or solo flute), biphonic texture (two melodic voices, such as a duo for bassoon and flute in which the bassoon plays a drone note and the flute plays the melody), polyphonic texture and homophonic texture (chords accompanying a melody).[citation needed]

The term musical form (or musical architecture) refers to the overall structure or plan of a piece of music,[76] and it describes the layout of a composition as divided into sections.[77] In the tenth edition of The Oxford Companion to Music, Percy Scholes defines musical form as "a series of strategies designed to find a successful mean between the opposite extremes of unrelieved repetition and unrelieved alteration."[78] According to Richard Middleton, musical form is "the shape or structure of the work." He describes it through difference: the distance moved from a repeat; the latter being the smallest difference. Difference is quantitative and qualitative: how far, and of what type, different. In many cases, form depends on statement and restatement, unity and variety, and contrast and connection.[79]

Musical analysis is the attempt to answer the questionhow does this music work? The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent, "analysis, as a pursuit in its own right, came to be established only in the late 19th century; its emergence as an approach and method can be traced back to the 1750s. However, it existed as a scholarly tool, albeit an auxiliary one, from the Middle Ages onwards."[80]Adolf Bernhard Marx was influential in formalising concepts about composition and music understanding towards the second half of the 19th century. The principle of analysis has been variously criticized, especially by composers, such as Edgard Varèse's claim that, "to explain by means of [analysis] is to decompose, to mutilate the spirit of a work".[81]

Schenkerian analysis is a method of musical analysis of tonal music based on the theories of Heinrich Schenker (1868–1935). The goal of a Schenkerian analysis is to interpret the underlying structure of a tonal work and to help reading the score according to that structure. The theory's basic tenets can be viewed as a way of defining tonality in music. A Schenkerian analysis of a passage of music shows hierarchical relationships among its pitches, and draws conclusions about the structure of the passage from this hierarchy. The analysis makes use of a specialized symbolic form of musical notation that Schenker devised to demonstrate various techniques of elaboration. The most fundamental concept of Schenker's theory of tonality may be that of tonal space.[82] The intervals between the notes of the tonic triad form a tonal space that is filled with passing and neighbour notes, producing new triads and new tonal spaces, open for further elaborations until the surface of the work (the score) is reached.

Although Schenker himself usually presents his analyses in the generative direction, starting from the fundamental structure (Ursatz) to reach the score, the practice of Schenkerian analysis more often is reductive, starting from the score and showing how it can be reduced to its fundamental structure. The graph of the Ursatz is arrhythmic, as is a strict-counterpoint cantus firmus exercise.[83] Even at intermediate levels of the reduction, rhythmic notation (open and closed noteheads, beams and flags) shows not rhythm but the hierarchical relationships between the pitch-events. Schenkerian analysis is subjective. There is no mechanical procedure involved and the analysis reflects the musical intuitions of the analyst.[84] The analysis represents a way of hearing (and reading) a piece of music.

Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his 1987 work, Generalized Musical Intervals and Transformations. The theory, which models musical transformations as elements of a mathematical group, can be used to analyze both tonal and atonal music. The goal of transformational theory is to change the focus from musical objects—such as the "C major chord" or "G major chord"—to relations between objects. Thus, instead of saying that a C major chord is followed by G major, a transformational theorist might say that the first chord has been "transformed" into the second by the "Dominant operation." (Symbolically, one might write "Dominant(C major) = G major.") While traditional musical set theory focuses on the makeup of musical objects, transformational theory focuses on the intervals or types of musical motion that can occur. According to Lewin's description of this change in emphasis, "[The transformational] attitude does not ask for some observed measure of extension between reified 'points'; rather it asks: 'If I am at s and wish to get to t, what characteristic gesture should I perform in order to arrive there?'"[85]

Musical expression is the art of playing or singing music with emotional communication. The elements of music that comprise expression include dynamic indications, such as forte or piano, phrasing, differing qualities of timbre and articulation, color, intensity, energy and excitement. All of these devices are at the service of the composer's intention and they can best be interpreted by the performer. A performer aims to elicit responses of sympathetic feeling in the audience, and to excite, calm or otherwise sway the audience's physical and emotional responses. In a great artist, one can feel that it is the soul that is speaking to the audience. In more modest performances, one can sometimes sense the soul of the composer in the absence of a heightened interpretation.[citation needed]

Expression can be closely related to breath, and the voice's natural ability to express feelings, sentiment, deep emotions.[clarification needed] Whether these can somehow be categorized is perhaps the realm of academics, who view expression as an element of musical performance which embodies a consistently recognizable emotion, ideally causing a sympathetic emotional response in its listeners.[88] The emotional content of musical expression is distinct from the emotional content of specific sounds (e.g., a startlingly-loud 'bang') and of learned associations (e.g., a national anthem), but can rarely be completely separated from its context.[citation needed]

The components of musical expression continue to be the subject of extensive and unresolved dispute.[89][90][91][92][93][94]

A music genre is a conventional category that identifies some pieces of music as belonging to a shared tradition or set of conventions.[95] It is to be distinguished from musical form and musical style, although in practice these terms are sometimes used interchangeably.[96][not in citation given]

Music can be divided into different genres in many different ways. The artistic nature of music means that these classifications are often subjective and controversial, and some genres may overlap. There are even varying academic definitions of the term genre itself. In his book Form in Tonal Music, Douglass M. Green distinguishes between genre and form. He lists madrigal, motet, canzona, ricercar, and dance as examples of genres from the Renaissance period. To further clarify the meaning of genre, Green writes, "Beethoven's Op. 61 and Mendelssohn's Op. 64 are identical in genre—both are violin concertos—but different in form. However, Mozart's Rondo for Piano, K. 511, and the Agnus Dei from his Mass, K. 317 are quite different in genre but happen to be similar in form."[97] Some, like Peter van der Merwe, treat the terms genre and style as the same, saying that genre should be defined as pieces of music that share a certain style or "basic musical language."[98]

Others, such as Allan F. Moore, state that genre and style are two separate terms, and that secondary characteristics such as subject matter can also differentiate between genres.[99] A music genre or subgenre may also be defined by the musical techniques, the style, the cultural context, and the content and spirit of the themes. Geographical origin is sometimes used to identify a music genre, though a single geographical category will often include a wide variety of subgenres. Timothy Laurie argues that since the early 1980s, "genre has graduated from being a subset of popular music studies to being an almost ubiquitous framework for constituting and evaluating musical research objects".[100]

Musical technique is the ability of instrumental and vocal musicians to exert optimal control of their instruments or vocal cords in order to produce the precise musical effects they desire. Improving one's technique generally entails practicing exercises that improve one's muscular sensitivity and agility. To improve their technique, musicians often practice fundamental patterns of notes such as the natural, minor, major, and chromatic scales, minor and major triads, dominant and diminished sevenths, formula patterns and arpeggios. For example, triads and sevenths teach how to play chords with accuracy and speed. Scales teach how to move quickly and gracefully from one note to another (usually by step). Arpeggios teach how to play broken chords over larger intervals. Many of these components of music are found in difficult compositions, for example, a large tuple chromatic scale is a very common element to classical and romantic era compositions as part of the end of a phrase.[citation needed]

Heinrich Schenker argued that musical technique's "most striking and distinctive characteristic" is repetition.[101] Works known as études (meaning "study") are also frequently used for the improvement of technique.

Music theorists sometimes use mathematics to understand music, and although music has no axiomatic foundation in modern mathematics, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".[102] The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.[103][104] There is a long history of examining the relationships between music and mathematics. Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,[105] the Pythagoreans (in particular Philolaus and Archytas)[106] of ancient Greece were the first researchers known to have investigated the expression of musical scales in terms of numerical ratios.

In the modern era, musical set theory uses the language of mathematical set theory in an elementary way to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as transposition and inversion, one can discover deep structures in the music. Operations such as transposition and inversion are called isometries because they preserve the intervals between tones in a set. Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the pitch classes in an equally tempered octave form an abelian group with 12 elements. It is possible to describe just intonation in terms of a free abelian group.[107][108]

"Integral serialism" or "total serialism" is the use of series for aspects such as duration, dynamics, and register as well as pitch. [110] Other terms, used especially in Europe to distinguish post–World War II serial music from twelve-tone music and its American extensions, are "general serialism" and "multiple serialism".[111]

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.[citation needed]

One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.[citation needed]

Music semiology (semiotics) is the study of signs as they pertain to music on a variety of levels. Following Roman Jakobson, Kofi Agawu adopts the idea of musical semiosis being introversive or extroversive—that is, musical signs within a text and without.[citation needed] "Topics," or various musical conventions (such as horn calls, dance forms, and styles), have been treated suggestively by Agawu, among others.[citation needed] The notion of gesture is beginning to play a large role in musico-semiotic enquiry.[citation needed]

"There are strong arguments that music inhabits a semiological realm which, on both ontogenetic and phylogenetic levels, has developmental priority over verbal language."[112]

Signs, meanings in music, happen essentially through the connotations of sounds, and through the social construction, appropriation and amplification of certain meanings associated with these connotations. The work of Philip Tagg (Ten Little Tunes,[full citation needed]Fernando the Flute,[full citation needed]Music’s Meanings[full citation needed]) provides one of the most complete and systematic analysis of the relation between musical structures and connotations in western and especially popular, television and film music. The work of Leonard Meyer in Style and Music[full citation needed] theorizes the relationship between ideologies and musical structures and the phenomena of style change, and focuses on romanticism as a case study.

Musical notation is the written or symbolized representation of music. This is most often achieved by the use of commonly understood graphic symbols and written verbal instructions and their abbreviations. In the 2000s, computer file formats have become important as well.[113] Spoken language and hand signs are also used to symbolically represent music, primarily in teaching.

In standard Western music notation, tones are represented graphically by symbols (notes) placed on a staff or staves, the vertical axis corresponding to pitch and the horizontal axis corresponding to time. Note head shapes, stems, flags, ties and dots are used to indicate duration. Additional symbols indicate keys, dynamics, accents, rests, etc. Verbal instructions from the conductor are often used to indicate tempo, technique, and other aspects.

In Western music, a range of different music notation systems are used. In Western Classical music, conductors use printed scores that show all of the instruments' parts and orchestra members read parts with their musical lines written out. In popular styles of music, much less of the music may be notated. A rock band may go into a recording session with just a handwritten chord chart indicating the song's chord progression using chord names (e.g., C major, d minor, G7, etc.). All of the chord voicings, rhythms and accompaniment figucres are improvised by the band members.

There are many systems of music notation from different cultures and different ages. Traditional Western notation evolved during the Middle Ages and remains an area of experimentation and innovation.[114]

The examples and perspective in this section deal primarily with the United States and do not represent a worldwide view of the subject. Please improve this article and discuss the issue on the talk page.(August 2015)

Music theorists typically hold a Ph.D in music theory. In the 1960s and 1970s, some music theorists obtained professor positions with an M.A. as their highest degree, but in the 2010s, the Ph.D is the standard minimum credential for tenure track professor positions.[citation needed] As part of their initial training, music theorists will typically complete a B.Mus or a B.A. in music (or a related field) and in many cases an M.A. in music theory. Some individuals apply directly from a bachelor's degree to a Ph.D, and in these cases, they may not receive an M.A. In the 2010s, given the increasingly interdisciplinary nature of university graduate programs, some applicants for music theory Ph.D programs may have academic training both in music and outside of music (e.g., a student may apply with a B.Mus and a Masters in Music Composition or Philosophy of Music).

Most music theorists work as instructors, lecturers or professors in colleges, universities or conservatories. The job market for tenure-track professor positions is very competitive.[vague] Applicants must hold a completed Ph.D or the equivalent degree and have a strong record of publishing in peer-reviewed journals. Some Ph.D-holding music theorists are only able to find insecure positions as sessional lecturers. The job tasks of a music theorist are the same as those of a professor in any other humanities discipline: teaching undergraduate and/or graduate classes in this area of specialization and, in many cases some general courses (such as Music Appreciation or Introduction to Music Theory), conducting research in this area of expertise, publishing research articles in peer-reviewed journals, authoring book chapters, books or textbooks, traveling to conferences to present papers and learn about research in the field, and, if the program includes a graduate school, supervising M.A. and Ph.D students and giving them guidance on the preparation of their theses and dissertations. Some music theory professors may take on senior administrative positions in their institution, such as Dean or Chair of the School of Music.

^Schenker described the concept in a paper titled Erläuterungen ("Elucidations"), which he published four times between 1924 and 1926: Der Tonwille[full citation needed] vol. 8–9, pp. 49–51, vol. 10, pp. 40–2; Das Meisterwerk in der Musik,[full citation needed] vol. 1, pp. 201–05; 2, p. 193-97. English translation, Der Tonwille,[full citation needed] vol. 2, p. 117-18 (the translation, although made from vols. 8–9 of the German original, gives as original pagination that of Das Meisterwerk[full citation needed] 1; the text is the same). The concept of tonal space is still present in Schenker (n.d., especially § 13), but less clearly than in the earlier presentation.

Charry, Eric (2000). Mande Music: Traditional and Modern Music of the Maninka and Mandinka of Western Africa. Chicago: University of Chicago Press. ISBN0-226-10161-4.Cite uses deprecated parameter |separator= (help)

Cheong, Kong F. (2012). "A Description of the Ceramic Musical Instruments Excavated from the North Group of Pacbitun, Belize". In Pacbitun Regional Archaeological Project, Report on the 2011 Field Season, edited by Terry G. Powis,[page needed]. Belmopan, Belize: Institute of Archaeology.

Chernoff, John (1981). African Rhythm and African Sensibility: Aesthetics and Social Action in African Musical Idioms. Chicago: University of Chicago Press. ISBN 978-0-226-10345-7.

Gerstner, Karl. 1964. Designing Programmes: Four Essays and an Introduction, with an introduction to the introduction by Paul Gredinger. English version by D. Q. Stephenson. Teufen, Switzerland: Arthur Niggli. Enlarged, new edition 1968.

Lu, Liancheng (2005). "The Eastern Zhou and the Growth of Regionalism". In The Formation of Chinese Civilization, edited by Sarah Allan,[page needed]. New Haven and London: Yale University Press. ISBN 978-0-300-09382-7.